Find the derivative of `cos (sin x^(2))" w.r.t.x. at x"=sqrt(pi/2)`.

Answer

To find the derivative of \( y = \cos(\sin(x^2)) \) with respect to \( x \) at \( x = \sqrt{\frac{\pi}{2}} \), we will use the chain rule of differentiation. Let's go through the steps one by one.

### Step 1: Define the function
Let
\[ y = \cos(\sin(x^2)) \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( \cos(u) \) is \( -\sin(u) \cdot \frac{du}{dx} \), where \( u = \sin(x^2) \).

Thus,
\[ \frac{dy}{dx} = -\sin(\sin(x^2)) \cdot \frac{d}{dx}(\sin(x^2)) \]

### Step 3: Differentiate \( \sin(x^2) \)
Next, we need to differentiate \( \sin(x^2) \). Again, we apply the chain rule:
\[ \frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot \frac{d}{dx}(x^2) \]
The derivative of \( x^2 \) is \( 2x \). Therefore,
\[ \frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x \]

### Step 4: Substitute back into the derivative
Now we can substitute this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -\sin(\sin(x^2)) \cdot (2x \cos(x^2)) \]

### Step 5: Evaluate at \( x = \sqrt{\frac{\pi}{2}} \)
Now we need to evaluate \( \frac{dy}{dx} \) at \( x = \sqrt{\frac{\pi}{2}} \):
1. First, calculate \( x^2 \):
\[ x^2 = \left(\sqrt{\frac{\pi}{2}}\right)^2 = \frac{\pi}{2} \]

2. Next, find \( \sin(x^2) \):
\[ \sin\left(\frac{\pi}{2}\right) = 1 \]

3. Now, substitute back into the derivative:
\[ \frac{dy}{dx} = -\sin(1) \cdot (2\sqrt{\frac{\pi}{2}} \cdot \cos\left(\frac{\pi}{2}\right)) \]

4. Since \( \cos\left(\frac{\pi}{2}\right) = 0 \), we have:
\[ \frac{dy}{dx} = -\sin(1) \cdot (2\sqrt{\frac{\pi}{2}} \cdot 0) = 0 \]

### Final Answer
Thus, the derivative of \( \cos(\sin(x^2)) \) with respect to \( x \) at \( x = \sqrt{\frac{\pi}{2}} \) is:
\[ \frac{dy}{dx} = 0 \]